OPERATIONS ON BITOPOLOGIES VIA IDEALS

Abstract

The concept ofbitopological space was initiated by J.C.Kelly [ 62] in

1963 . After the publication of Kelly's paper many toplogiests have shown interest in the study of such spaces . The problem of defining bitopological compactness has been considered by several authors in particular, Fletcher, Hoyle and Patty [38] , Kim [64], Birsan [22] ,Swart [112] and Reilly (103].

In 1982 , Munshi and Bassan [87] have introduced the concept of S­ compact spaces . Mashhour , et . al. have defined the concept of pairwise S - compactness (82] . In other paper Mashhour , et .al .have study the notion of pairwise - S- closed spaces [ 97] . Swart [112] , has introduced the concept of pairwise - C - compactness . Reilly [102] , has introduced the concept of pairwise - Lindelof spaces . Abd El-Monsef, et. al. have introduced the concepts of pairwise - S - Lindel()f spaces , pairwise - M - S - Lindelof spaces , pairwise - S - C compact spaces , pairwise - S - closed spaces and pairwise- S- closed sets [12] .

The genesis of concept of ideals in general topological spaces is treated in the classic text by Kuratawski in 1966 [66] .Also, several authors have interested in this line of study and therefore some sorts of ideals arise as one goes further in mathematics such as the ideal of finite subsets ofX, the ideal of nowhere dense sets and the ideal of meager sets . Different types of operators in terms of ideals , compactness via an ideal , sets , functions and other concepts were investigated by many topologists . The concept of the set operator ( )• : P(X) -+ P(X) had been introduced by